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Theorem symgvalstructOLD 19102
Description: Obsolete proof of symgvalstruct 19101 as of 6-Nov-2024. The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
symgvalstructOLD.g 𝐺 = (SymGrp‘𝐴)
symgvalstructOLD.b 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
symgvalstructOLD.m 𝑀 = (𝐴m 𝐴)
symgvalstructOLD.p + = (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔))
symgvalstructOLD.j 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
Assertion
Ref Expression
symgvalstructOLD (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Distinct variable groups:   𝐴,𝑓,𝑔   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝑥,𝐽   𝑓,𝑀,𝑔   𝑥,𝑉   𝑥, +
Allowed substitution hints:   𝐵(𝑓,𝑔)   + (𝑓,𝑔)   𝐺(𝑓,𝑔)   𝐽(𝑓,𝑔)   𝑀(𝑥)   𝑉(𝑓,𝑔)

Proof of Theorem symgvalstructOLD
StepHypRef Expression
1 hashv01gt1 14161 . 2 (𝐴𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2 hasheq0 14179 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3 0symgefmndeq 19098 . . . . . . . . 9 (EndoFMnd‘∅) = (SymGrp‘∅)
43eqcomi 2745 . . . . . . . 8 (SymGrp‘∅) = (EndoFMnd‘∅)
5 symgvalstructOLD.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐴)
6 fveq2 6826 . . . . . . . . 9 (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
75, 6eqtrid 2788 . . . . . . . 8 (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8 fveq2 6826 . . . . . . . 8 (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
94, 7, 83eqtr4a 2802 . . . . . . 7 (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
109adantl 482 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11 eqid 2736 . . . . . . . 8 (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
12 symgvalstructOLD.m . . . . . . . 8 𝑀 = (𝐴m 𝐴)
13 symgvalstructOLD.p . . . . . . . 8 + = (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔))
14 symgvalstructOLD.j . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
1511, 12, 13, 14efmnd 18606 . . . . . . 7 (𝐴𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
1615adantr 481 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17 0map0sn0 8745 . . . . . . . . . . 11 (∅ ↑m ∅) = {∅}
18 id 22 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
1918, 18oveq12d 7356 . . . . . . . . . . 11 (𝐴 = ∅ → (𝐴m 𝐴) = (∅ ↑m ∅))
20 symgvalstructOLD.b . . . . . . . . . . . 12 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
217fveq2d 6830 . . . . . . . . . . . . 13 (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22 eqid 2736 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
235, 22symgbas 19075 . . . . . . . . . . . . 13 (Base‘𝐺) = {𝑥𝑥:𝐴1-1-onto𝐴}
24 symgbas0 19093 . . . . . . . . . . . . 13 (Base‘(SymGrp‘∅)) = {∅}
2521, 23, 243eqtr3g 2799 . . . . . . . . . . . 12 (𝐴 = ∅ → {𝑥𝑥:𝐴1-1-onto𝐴} = {∅})
2620, 25eqtrid 2788 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐵 = {∅})
2717, 19, 263eqtr4a 2802 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴m 𝐴) = 𝐵)
2827adantl 482 . . . . . . . . 9 ((𝐴𝑉𝐴 = ∅) → (𝐴m 𝐴) = 𝐵)
2912, 28eqtrid 2788 . . . . . . . 8 ((𝐴𝑉𝐴 = ∅) → 𝑀 = 𝐵)
3029opeq2d 4825 . . . . . . 7 ((𝐴𝑉𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
3130tpeq1d 4694 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3210, 16, 313eqtrd 2780 . . . . 5 ((𝐴𝑉𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3332ex 413 . . . 4 (𝐴𝑉 → (𝐴 = ∅ → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
342, 33sylbid 239 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35 hash1snb 14235 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36 snex 5377 . . . . . . . 8 {𝑥} ∈ V
37 eleq1 2824 . . . . . . . 8 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
3836, 37mpbiri 257 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 ∈ V)
3911, 12, 13, 14efmnd 18606 . . . . . . 7 (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
4038, 39syl 17 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
41 snsymgefmndeq 19099 . . . . . . 7 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
4241, 5eqtr4di 2794 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
4342fveq2d 6830 . . . . . . . . 9 (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44 eqid 2736 . . . . . . . . . . 11 (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
4511, 44efmndbas 18607 . . . . . . . . . 10 (Base‘(EndoFMnd‘𝐴)) = (𝐴m 𝐴)
4645, 12eqtr4i 2767 . . . . . . . . 9 (Base‘(EndoFMnd‘𝐴)) = 𝑀
4723, 20eqtr4i 2767 . . . . . . . . 9 (Base‘𝐺) = 𝐵
4843, 46, 473eqtr3g 2799 . . . . . . . 8 (𝐴 = {𝑥} → 𝑀 = 𝐵)
4948opeq2d 4825 . . . . . . 7 (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
5049tpeq1d 4694 . . . . . 6 (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5140, 42, 503eqtr3d 2784 . . . . 5 (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5251exlimiv 1932 . . . 4 (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5335, 52syl6bi 252 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54 ssnpss 4051 . . . . . . 7 ((𝐴m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴m 𝐴))
5511, 5symgpssefmnd 19100 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5620, 23eqtr4i 2767 . . . . . . . . 9 𝐵 = (Base‘𝐺)
5745eqcomi 2745 . . . . . . . . 9 (𝐴m 𝐴) = (Base‘(EndoFMnd‘𝐴))
5856, 57psseq12i 4039 . . . . . . . 8 (𝐵 ⊊ (𝐴m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5955, 58sylibr 233 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊊ (𝐴m 𝐴))
6054, 59nsyl3 138 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ¬ (𝐴m 𝐴) ⊆ 𝐵)
61 fvexd 6841 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62 f1osetex 8719 . . . . . . . 8 {𝑥𝑥:𝐴1-1-onto𝐴} ∈ V
6320, 62eqeltri 2833 . . . . . . 7 𝐵 ∈ V
6463a1i 11 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
655, 20symgval 19073 . . . . . . 7 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
6665, 57ressval2 17044 . . . . . 6 ((¬ (𝐴m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
6760, 61, 64, 66syl3anc 1370 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
68 ovex 7371 . . . . . . 7 (𝐴m 𝐴) ∈ V
6968inex2 5263 . . . . . 6 (𝐵 ∩ (𝐴m 𝐴)) ∈ V
70 setsval 16966 . . . . . 6 (((EndoFMnd‘𝐴) ∈ V ∧ (𝐵 ∩ (𝐴m 𝐴)) ∈ V) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
7161, 69, 70sylancl 586 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
7215adantr 481 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
7372reseq1d 5923 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
7473uneq1d 4110 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
75 eqidd 2737 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76 fvexd 6841 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77 fvexd 6841 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
7812, 68eqeltri 2833 . . . . . . . . . . 11 𝑀 ∈ V
7978, 78mpoex 7989 . . . . . . . . . 10 (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔)) ∈ V
8013, 79eqeltri 2833 . . . . . . . . 9 + ∈ V
8180a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
8214fvexi 6840 . . . . . . . . 9 𝐽 ∈ V
8382a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84 basendxnplusgndx 17090 . . . . . . . . . 10 (Base‘ndx) ≠ (+g‘ndx)
8584necomi 2995 . . . . . . . . 9 (+g‘ndx) ≠ (Base‘ndx)
8685a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87 tsetndx 17160 . . . . . . . . . 10 (TopSet‘ndx) = 9
88 1re 11077 . . . . . . . . . . . 12 1 ∈ ℝ
89 1lt9 12281 . . . . . . . . . . . 12 1 < 9
9088, 89gtneii 11189 . . . . . . . . . . 11 9 ≠ 1
91 df-base 17011 . . . . . . . . . . . 12 Base = Slot 1
92 1nn 12086 . . . . . . . . . . . 12 1 ∈ ℕ
9391, 92ndxarg 16995 . . . . . . . . . . 11 (Base‘ndx) = 1
9490, 93neeqtrri 3014 . . . . . . . . . 10 9 ≠ (Base‘ndx)
9587, 94eqnetri 3011 . . . . . . . . 9 (TopSet‘ndx) ≠ (Base‘ndx)
9695a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
9775, 76, 77, 81, 83, 86, 96tpres 7133 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9897uneq1d 4110 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
99 uncom 4101 . . . . . . . 8 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
100 tpass 4701 . . . . . . . 8 {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10199, 100eqtr4i 2767 . . . . . . 7 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
1025, 56symgbasmap 19081 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ (𝐴m 𝐴))
103102a1i 11 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥𝐵𝑥 ∈ (𝐴m 𝐴)))
104103ssrdv 3938 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴m 𝐴))
105 df-ss 3915 . . . . . . . . . 10 (𝐵 ⊆ (𝐴m 𝐴) ↔ (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
106104, 105sylib 217 . . . . . . . . 9 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
107106opeq2d 4825 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
108107tpeq1d 4694 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
109101, 108eqtrid 2788 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
11074, 98, 1093eqtrd 2780 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
11167, 71, 1103eqtrd 2780 . . . 4 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
112111ex 413 . . 3 (𝐴𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
11334, 53, 1123jaod 1427 . 2 (𝐴𝑉 → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
1141, 113mpd 15 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3o 1085   = wceq 1540  wex 1780  wcel 2105  {cab 2713  wne 2940  Vcvv 3441  cdif 3895  cun 3896  cin 3897  wss 3898  wpss 3899  c0 4270  𝒫 cpw 4548  {csn 4574  {cpr 4576  {ctp 4578  cop 4580   class class class wbr 5093   × cxp 5619  cres 5623  ccom 5625  1-1-ontowf1o 6479  cfv 6480  (class class class)co 7338  cmpo 7340  m cmap 8687  0cc0 10973  1c1 10974   < clt 11111  9c9 12137  chash 14146   sSet csts 16962  ndxcnx 16992  Basecbs 17010  +gcplusg 17060  TopSetcts 17066  tcpt 17247  EndoFMndcefmnd 18604  SymGrpcsymg 19071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-cnex 11029  ax-resscn 11030  ax-1cn 11031  ax-icn 11032  ax-addcl 11033  ax-addrcl 11034  ax-mulcl 11035  ax-mulrcl 11036  ax-mulcom 11037  ax-addass 11038  ax-mulass 11039  ax-distr 11040  ax-i2m1 11041  ax-1ne0 11042  ax-1rid 11043  ax-rnegex 11044  ax-rrecex 11045  ax-cnre 11046  ax-pre-lttri 11047  ax-pre-lttrn 11048  ax-pre-ltadd 11049  ax-pre-mulgt0 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-om 7782  df-1st 7900  df-2nd 7901  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-oadd 8372  df-er 8570  df-map 8689  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-dju 9759  df-card 9797  df-pnf 11113  df-mnf 11114  df-xr 11115  df-ltxr 11116  df-le 11117  df-sub 11309  df-neg 11310  df-nn 12076  df-2 12138  df-3 12139  df-4 12140  df-5 12141  df-6 12142  df-7 12143  df-8 12144  df-9 12145  df-n0 12336  df-xnn0 12408  df-z 12422  df-uz 12685  df-fz 13342  df-hash 14147  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-tset 17079  df-efmnd 18605  df-symg 19072
This theorem is referenced by: (None)
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